The second part follows from Itô's lemma and is standard. For the first part we need to combine the argument of [1]} and [2]}. From [2]} Theorem 2.5 we know that this statement is equivalent to the existence of a solution to the so-called martingale problem and from [1]} Chapter 4, we know that existence of a solution to the martingale problem is equivalent to the existence of a weak solution to the SDE.
